ODE-Determining whether differential and all partial derivates are continuous

Please help!
I have been given the condition:
U is a ball of radius r>0 in R^n, it is centred at a.
x is an element in U given length (x-a)<r
The ordinary differential equation:
dx/dt = v(t,x),
where x=(x1, x2, ..., xn)^T is an element in R^n.
If U is continuous and all partial derivatives are continuous in U. Let |t-t0)<T, then dx/dt=v(t,x) has a unique solution satisfying x(t0)=a defined on interval J which is part of R.
Determine whether the following equations satisfy the above condition:
(a): dx/dt=Ax^(1/2), where A is a constant
(b): dx/dt=x^2

Any help on proving whether the equations satisfy this condition will be VERY appreciated, or even just some information on the topic/process that will be helpful!